How to mask off part of image plane
Materials only solution

Create a cube
name it cookie cutter  (Optional)
Give it this material:

Apply this material:

to the background plane, and position cookie cutter to make the hole.

Note: When the camera looks through the cookie cutter at the plane, the plane will be transparent, so make the cookie cutter narrow. It may also be affected by other objects, so be careful.
Texture + Material Solution

Apply this material to the background plane

In the top image node, give it a black and white image, where it is white, there will be a hole.

Note: This is very robust, but can be a huge pain if you start moving your objects.